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Algebra Difficulty 3.9 AMC 10/12 Find the answer China

Suppose mm is a real number, and complex numbers z1=1+2iz_1 = 1 + 2i, z2=m+3iz_2 = m + 3i, where ii is the imaginary unit. If z1⋅z2ˉz_1 \cdot \bar{z_2} is purely imaginary, then the value of ∣z1+z2∣|z_1 + z_2| is ______.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Since z1⋅z2ˉ=(1+2i)(m−3i)=m+6+(2m−3)iz_1 \cdot \bar{z_2} = (1 + 2i)(m - 3i) = m + 6 + (2m - 3)i is purely imaginary, we get m=−6m = -6. Therefore, ∣z1+z2∣=∣−5+5i∣=52|z_1 + z_2| = |-5 + 5i| = 5\sqrt{2}.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.